Clinic Scheduling Models with Overbooking for Patients with Heterogeneous No-show Probabilities

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1 Clinic Scheduling Models with Overbooking for Patients with Heterogeneous No-show Probabilities Bo Zeng, Ji Lin and Mark Lawley {bzeng, lin35, malawley}@purdue.edu Weldon School of Biomedical Engineering, Purdue University, West Lafayette, IN Ayten Turkcan, e-enterprise Center, Purdue University, West Lafayette, IN aturkcan@purdue.edu February 26, Abstract Clinical overbooking is intended to reduce the negative impact of patient no-shows on clinic operations and performance. In this paper, we study the clinical scheduling problem with overbooking for heterogeneous patients, i.e. patients who have different no-show probabilities. We consider the objective of maximizing expected profit, which includes revenue from patients and costs associated with patient waiting times and physician overtime. We show that the objective function with homogeneous patients, i.e. patients with the same no-show probability, is multimodular. We also show that this property does not hold when patients are heterogeneous. We identify properties of an optimal schedule with heterogeneous patients and propose a local search algorithm to find local optimal schedules. Then, we extend our results to sequential scheduling and propose two sequential scheduling procedures. Finally, we perform a set of numerical experiments and provide managerial insights for health care practitioners. Key Wordsclinical scheduling, overbooking, patient no-show, multimodularity Introduction The majority of patient care in the U.S. (80 90% [2, 4]) is provided by outpatient clinics. Clinic operations are typically driven by appointment schedules, and appointment scheduling is often cited by clinic managers as a major opportunity for improvement. Cayirli and Veral [3] provide a comprehensive review of research in outpatient appointment scheduling. They state that most analytical research does not consider factors such as patient no-shows, walk-ins and emergency. Nevertheless, these factors have significant adverse effect on operational efficiency, total revenue and patient satisfaction. Among these factors, patient no-show is of particular concern because it wastes the available capacity of valuable resources (physician, staff, equipment) and limits clinic access to the patient population. Cayirli and Veral [3] mention that no-show rates are 5-30%. However, Rust et al. [17] report that for some health care settings, such as public pediatric clinics, no-show rates can reach 80%. To reduce the negative impact of patient no-show, clinic schedulers often overbook. However, naive overbooking can lead to longer patient waiting times, clinic overtime, and deteriorating outcomes for patients who leave without being seen [8, 18]. Thus, modeling and analysis must be used to develop a scheduling methodology that properly balances these competing objectives. Supported by NSF Grant CMMI

2 An ideal overbooking model depends on four characteristics. The first is a valid patient no-show description that captures the real pattern of patient behavior. The second is the underlying service model that reflects the operational dynamics of the clinic. The third is an objective function that reflects the performance concern of clinic managers. And the last is an efficient algorithm that can generate schedules of desired quality in a timely fashion. We give a brief review of existing clinic overbooking models categorized according to these four characteristics. We also include some studies that do not explicitly consider overbooking but that can be used to obtain overbooked schedules. No-show probabilities can be correlated to factors such as reservation lead time and patient demographics, see Garuda et al. [5] for details. Even though no-show usually differs by patient, almost all overbooking studies assume that patients are homogeneous, i.e. all patients have the same no-show probability. Liu and Liu [12], Laganga and Lawrence [10, 11], Kim and Giachetti [8] and Kaandorp and Koole [7] consider a single no-show rate for all patients in their models. Muthuraman and Lawley [15] provide an exception by explicitly modeling different no-show probabilities. With clinic dynamics, most researchers develop single server models (a schedule is created for a single physician). Kim and Giachetti [8] and Laganga and Lawrence [10, 11] assume that the service times of patients are deterministic while Kaandorp and Koole [7] and Muthuraman and Lawley [15] use queuing based models with exponential service times. Liu and Liu [12] consider a model for multiple servers and investigate service times with exponential and general distributions. The performance criteria considered in appointment scheduling models includes revenue from patients, patient waiting time/cost, physician overtime/cost and physician idle time/cost. Kim and Giachetti [8] consider expected revenue and physician overtime. Since the cost of patient waiting time is not included in their model, all patients are assumed to arrive at the beginning of a clinic session, which implies a single block scheduling model. Laganga and Lawrence [11] consider revenue from patients and costs of patient waiting time and physician overtime, in both linear and quadratic objective functions. Kaandorp and Koole [7] explicitly include the cost of physician idle time, as do Liu and Liu [12]. Muthuraman and Lawley [15] consider revenue from patients and costs from patients waiting time and clinic overtime. In most cases, computing the optimal schedule is computationally intractable and thus most scheduling algorithms are heuristics or simulation based methods [7, 8, 10, 11, 12]. The research by Kaandorp and Koole [7] is of special interest because they show that their model is multimodular. Multimodularity is a property of functions in discrete space, similar to convexity in continuous space, which guarantees that a locally optimal solution is also globally optimal. In contrast to the work just mentioned, Muthuraman and Lawley [15] consider sequential scheduling in which the schedule is constructed as patients seeking appointments call clinic schedulers. Patients must be given their appointment time before the call ends, and thus once a patient appointment is added to the schedule, it is typically not feasible to alter the time. In this case, the set of patients to be scheduled is not initially known and deciding when a schedule is complete becomes a problem. Although the authors provide a scheduling algorithm and derive optimal stopping criteria, the optimal sequential schedule is not characterized. In this study, we derive some properties of an optimal schedule, which can be used to design better algorithms. The existing studies do not adequately address the question of how the scheduling problem for heterogeneous patients is different from that of homogeneous patients and whether modeling the heterogeneous nature of patient no-show can lead to superior schedules, particularly in a sequential setting where schedules have to be constructed as patients call-in. Even though different no-show probabilities are taken into account in [15], the patients are treated equally while scheduling. The decision to accept (or reject) a patient is given by only looking at the increase (or decrease) in the objective function. However, scheduling patients with high no-show probabilities leads to higher variabilities in daily workload of clinics. In this study, we also investigate the effect of variability in no-show rates on the quality of the resulting schedules. The remainder of the paper is structured as follows. Section 2 provides the notation and defines the basic problem. In Section 3, we study the structure of the optimization model and prove that it is not multimodular in general. In Section 4, after deriving some important properties of optimal schedules, we propose a local search algorithm to obtain a good schedule and discuss its extension to sequential 2

3 scheduling settings. In Section 5, we present a computational study to compare the proposed algorithms with the existing methods. Section 6 concludes with some managerial insights that can improve patients scheduling and clinic performance in practice. 2 Problem Definition We first state our assumptions and then present required notation. We assume that all patient arrivals to the clinic are scheduled (no walk-ins) and that the patient population can be partitioned into categories based on no-show probability. We further assume that the clinic day is partitioned into a set of time slots and that patient appointment times coincide with the beginning of a slot. We also assume that all arriving patients are punctual and that patients are served according to a first-come-first-serve protocol. Finally, we assume that service times are exponential and that they are independent and identically distributed across patients. Notation is as follows: I set of slots i slot set index, i {1,..., } t i length of slot i J set of patient types based on no-show j patient type, j {1,..., J } X i number of patients arriving at start of slot i Y i number of patients in the system at the end of slot i, overflow from slot i to slot i + 1 L i number of service completions in slot i λ service rate c i unit overflow cost from slot i to slot i + 1 (i ), c i 0 c I unit overflow cost for i =, unit overtime cost, typically c > c i r revenue per patient, r > 0 p j probability that patient of type j arrives as scheduled, (p 1 > > p J ) n j number of patients of type j S a schedule ( Z J ) i,j unit matrix such that cell i, j is 1, all others 0 F(S) objective function value of S R(S) overflow matrix of schedule S Q(S) arrival matrix of schedule S a b random variables a and b are iid For a given set of heterogeneous patients, we formulate the following overbooking model to obtain an optimal schedule S that maximizes the expected total profit. max F(S) =r i I E[X i ] i I c i E[Y i ] s.t. i I S i,j n j (1) S i,j Z i I, j J We note that X i + Y i 1 is the number of patients in the system at the beginning of slot i and that the number of patients in the system at the end of slot i is given as: Y i = max{y i 1 + X i L i, 0}. (2) To compute probabilities for X i and Y i, Muthuraman and Lawley [15] introduce two matrices, an arrival matrix [Q i,l ] such that Q i,l is the probability that l patients arrive at the beginning of slot i, and 3

4 an overflow matrix [R i,k ] such that R i,k is the probability that k patients overflow from slot i to slot i + 1. These are computed as follows: Q i,l (S) = Pr(X i = l) = π Ω j J S i,j! π j!(s i,j π j )! pπj j (1 p j) Si,j πj, where π = {π 1,..., π J } with π j Z + for j J, j J π j = l and Ω is the set of all such vectors. (1 F Li (l + k))q i,l R i 1,k if m = 0 R i,m (S) = l k f Li (l + k m)q i,l R i 1,k if m 1 l k (3) f Li (k) = F Li (k) = e (λt i) k λti k! k 1 f Li ( k). k= Given these equations, we can compute E[X i ] = l iq i,l and E[Y i ] = k kr i,k. Typically, optimization problems such as (1) arising from appointment service systems are very difficult to solve since the objective functions are nonlinear and decision variables are discrete. However, it has recently been shown that if the objective function is multimodular over Z n, a property similar to convexity in R n, and constraints are simple upper or lower bound constraints, a well-defined local search algorithm can be used to obtain (global) optimal solutions, see Hajek [6], Altman et al. [1] and Koole and van der Sluis [9]. Based on these results, Kaandorp and Koole [7] prove that their scheduling model for homogeneous patients is multimodular and implement a local search algorithm to obtain an optimal schedule. As a natural extension, it is important to see whether our overbooking model is multimodular. If so, we can use the results to obtain an optimal scheduling method, and if not we are justified in seeking heuristic approaches. Section 3 addresses this problem. 3 Structure of the Overbooking Scheduling Model In this section, we investigate the multimodularity of the scheduling model given in (1). As an aid to the reader, we make the following informal note about multimodularity before providing the definition. Let f be function on Z m. When we join the integer points of f by lines, we obtain a new function g on R m. g is convex if and if f is multimodular. This implies that a local optimum is also a global optimum. More formally, let e i be the i th standard unit vector of R m. Then, we define a set of vectors Γ = { v 0,..., v m } Z m such that v 0 = e 1, v i = e i e i+1, for i = 1,...,m 1 and v m = e m. Definition 1. A function f : Z m R is multimodular if for all x Z m, u, v Γ, u v, f( x + u) + f( x + v) f( x) + f( x + u + v). (4) 130 Because of the connection between multimodular and convex functions, Koole and van der Sluis [9] 131 propose a local search algorithm that searches all the neighbors of a particular point x in the form x v U v where U is a subset of Γ. They show that this local search will lead to an optimal solution 133 of f. Later, Kaandorp and Koole [7] use this concept to obtain an optimal schedule for their scheduling 134 model. In this section,we use the following equivalent form f( x + u) f( x) f( x + v + u) f( x + v) (5) to verify the multimodularity of a function. We note that (5) can be interpreted as the improvement from perturbing x by u is greater or equal to that from perturbing x + v by u. Because u and v are 4

5 closely related to the unit vectors e i for some i, we first derive some result on the improvement of f obtained from perturbing x by e i. This result will be frequently used in this section to help us simplify the proof of multimodularity. Proposition 1. For a given schedule S 0, we have F(S 0 + i,j 1 ) F(S 0 ) F(S 0 + i,j 2 ) F(S 0 ) = p j 1 p j2 (6) for all i I and j 1, j 2 J. Proof. Assume that W is a patient with no-show probability p j1 being added to slot i in schedule S 0 such that the schedule is updated by S 1 = S 0 + i,j 1. We use Xi 0 and Yi 0 to denote the number of arrivals in slot i and the size of overflow from slot i, respectively, for schedule S 0. Then, we define Xi 1 and Yi 1 for S 1 similarly. Also, we introduce P i (i ) to be the conditional probability that the arrival of patient W increases the overflow from slot i to i + 1 by 1. Let W denote the arrival of patient W. Then, from (1), on the condition of W, we have F(S 1 ) F(S 0 ) =rp j1 + i I c i E[Y 1 i Y 0 i ] =rp j1 (1 p j1 ) i I =p j1 (r i I c i P i (i )). 0 p j1 i I c i E[Y 1 i Y 0 i W] (7) It can be easily seen that if P i (i ) is independent of p j1 for all i, we have F(S 0 + i,j 2 ) F(S 0 ) = p j2 (r i I c ip i (i )). Then, the conclusion follows. In fact, we observe that the only situation where the arrival of patient W leads to one more patient overflowing from slot i is the case where for each slot k such that i k i, the number of patients served is less than or equal to the number patients in slot k in schedule S 0. So, we have { i P i (i ) = k=i Pr(L k Xk 0 + Y k 1 0 ) if i i 0 otherwise. (8) Clearly, both P i (i ) and r i I c ip i (i ) are independent of the no-show probability of patient W. The desired results follows. Because the result of Proposition 1 is about the value difference from unit changes, we call the result of (7) the local perturbation of a given schedule. Next, we show that the concept of local perturbation can help us simplify the proof of multimodularity significantly as compared to that of Kaandorp and Koole [7]. Because (1) is a maximization problem, we use F to denote F and verify that F is multimodular. When J = 1, we use S i instead of S i,j and use p to denote the patient show-up probability in our derivation. Theorem 2. When J = 1, F is a multimodular function over Z. Proof. Because for the simple cases where u or v is e 1 or e, (5) can easily be proven using the argument similar to the following, we focus on the general case where neither u and v are standard unit vectors. Without loss of generality, we let u = e k e k+1 and v = e l e l+1 such that 1 k < l 1. To make use of our local perturbation, for any particular schedule S, we define two base schedules S L and S R for LHS and RHS of (5) as S L = S e k+1 (9) and S R = S e k+1 + e l e l+1 (10) 5

6 with S k+1 1 and S l+1 1. By using S L and S R, both LHS and RHS can be interpreted as the difference of two local perturbations. Correspondingly, we use Xi L, XR i, Y i L, Y i R to denote the number of arrivals in slot i and the number of overflows from slot i in S L and S R, respectively. We also use Pi L(i 0) and Pi R(i 0) to denote the overflow effect from adding one more patient to slot i 0 in schedule S L and S R on the condition of this patient s arrival. From (8), we have (11) for LHS of (5) F(S + u) F(S) = (F(S + u) F(S L )) (F(S) F(S L )) = p i=k c i P L i (k) p c i P L i (k + 1) = p{c k P L k (k) + Pr(L k X L k + Y L k 1) c i i h=k+1 Pr(L h x L h + Y L h 1)} = p{c k Pk L (k) + (Pr(L k Xk L + Y k 1 L ) 1) c i i h=k+1 Pr(L h x L h + Y L h 1) c i Pi L (k + 1)}. (11) Note that, in the first equality of (11), the first sum evaluates overflow characteristics when the patient is added to slot k and the second sum evaluates overflow characteristics when the patient is added to slot k + 1. Similarly, we have (12) for RHS of (5). F(S + u + v) F(S + v) = (F(S + u) F(S R )) (F(S) F(S r )) = p c i Pi R (k) p i=k c i Pi R (k + 1) = p{c k Pk R (k) + (Pr(L k Xk R + Y k 1 R ) 1) c i Pi R (k + 1)}. (12) Because S L and S R have same number of patients per slot up to and including slot l 1 k, we have Pk L(k) = P k R(k) = Pr(L k Xk L + Y k 1 L ) = Pr(L k Xk R + Y k 1 R ). Also, because Pr(L k Xk L + Y k 1 L ) 1 0, it is sufficient to show that c i P L i (k + 1) c i P R i (k + 1) Observe that S R can be obtained from S L by reassigning one patient who is in slot l+1 to slot l. Let W be a such patient. Then, we can compare (11) and (12) conditioned on the arrival of W, W. Clearly, if W, we have (11) and (12) are same. If W, we have Xl R Xl L + 1, Xl+1 R XL l 1, Yi R Yi L for i = k,..., l 1 and Xi R Xi L for i l, l + 1. Furthermore, on the condition of W, we also have Pr(L l X R l + Y R l 1 ) = Pr(L l X L l + Y L l 1 + 1) = Pr(L l X L l + Y L l 1 ) + Pr(L l = X L l + Y L l 1 + 1). 186 Next, we compare the dynamics of queuing model in S L and S R in the case where L l Xl L + Yl 1 L 187 Because Xl R Xl L + 1, Yl 1 R l 1 L l Xl L + Yl 1 L R, we have Yl Yl L + 1. Also, because Xl+1 R XL l+1 1, we have XR l+1 + Y l+1 R l+1 + Y l+1 L R and therefore Yl+1 l+1 L. From the fact that 189 Xj R Xj L for j l + 1, we have Xj R + Y j 1 R j + Y j 1 L for j = l + 1,...,. Clearly, from slot l to slot, the queuing dynamics in schedule SR and SL are identical, as shown in Figure 1. 6

7 Figure 1: Dynamics of Slot Queuing System in S R and S L Conditioned on W and L l X L l + Y L l Let H R i = i j=l+1 Pr(L j X R j X R j + Y R j 1 ) given L l = X L l + Y L l additional overflow from slot i and Hi R 194 flow from slot i. Further, Hi R 195 consequence, we obtain l 1 c i Pi R (k + 1) = = + Yj 1 R ) given L l Xl L + Yl 1 L and HR i j=l+1 Pr(L j + 1 for i l + 1. HR i represents the probability that W causes represents the probability that W causes no additional overflow Pr(L l X L l + Y L l 1 ) = i j=l Pr(L j X L j + Y L j 1 = i ) for for i l + 1. As a c i P L i (k + 1) + c lp L l 1 (k + 1)(Pr(L l X L l + Y L l 1 ) + Pr(L l = X L l + Y L + 1)) + Pl 1 L (k + 1)( l 1 i=l+1 c i Hi R Pr(L l Xl L + Yl 1 L ) + i=l+1 c i P L i (k + 1) + c l P L l 1(k + 1)Pr(L l X L l + Y L l 1)+ Pl 1 L (k + 1)Pr(L l Xl L + Yl 1 L ) c i Pi L (k + 1). i=l+1 c i H R i c i H R i Pr(L l = X L l + Y L + 1)) 196 The inequality follows from the fact that c i 0 for i I and probabilities are non-zero. The last equality follows from the definition of Pi L 197 (k) in (8). Since (1) is multimodular for J = 1, we can apply the local search method by Koole and van der (13) Sluis [9] to obtain an optimal schedule. However, for the general case where J 2, we have the following result. Note that we express a schedule S in the form of a vector, [S 1,1,..., S 1, J,..., S,1,..., S J ] Z J. Theorem 3. The function F is not multimodular over Z J for J 2. Proof. It is sufficient to show that for some u, v Γ, (5) does not hold. Let u = e l e l+1 and v = e l+k J e l+k J +1 for some l, k such that 1 l, l + k J + 1 J. Denote j 0 = l 204 J + 1. Then, 205 we observe that the operation corresponding to u ( v, respectively) is to move one patient of p (p j0+k, j0 206 respectively) from slot i = l + 1 (j 0 1) J and to slot i 0. 7

8 Similar to our proof for Theorem 2, for a particular schedule S, we define two base schedule S L and S R for LHS and RHS of (5) as S L = S e l+1 (14) and S R = S e l+1 + e l+k J e l+k J +1. (15) We also use Xi L, XR i, Y i L, and Y i R to denote arrival and overflow in S L and S R 210 respectively. Comparing S L and S R, we observe that Si R 211 = 0,j 0+k SL i + 1, 0,j 0+k SR i = 0+1,j 0+k SL i 1 and 0+1,j 0+k SR i,j = SL i,j for all other (i, j). We can easily see that S R the scheduling resulting when we reassign one patient of type j 0 + k from slot i to slot i 0 in schedule S L. Let W be a such patient. Again, we use Pi L(i 0) and Pi R(i 0) to denote the overflow effect from adding one more patient to slot i 0 in schedule S L and S R conditioned on W. If W, S R = S L. So, we need only consider the case where W shows up. Similar to the proof of Theorem 2, for LHS of (5), we have LHS = F(S + u) F(S) =c i0 Pr(L i0 X L i 0 + Y L i 0 1 ) For RHS of (5), we have + (Pr(L i0 Xi L 0 + Yi L ) 1)( 0 1 i=i 0+1 c i i k=i 0+1 Pr(L k X L k + Y L k 1 ) (16) RHS = F(S + u + v) F(S + v) =c i0 Pr(L i0 X R i 0 + Y R i 0 1 ) + (Pr(L i0 X R i 0 + Y R i 0 1) 1)( i=i 0+1 c i i k=i 0+1 Pr(L k X R k + Y R k 1) Next, we compare the value of (16) and (17) conditioned on the physician s performance in slot i 0, i.e. L i0. Note that under W, we have Xi R 0 + Yi R 0 Xi L 0 + Yi L and Xi R 0+1 XL i Case (i) L i0 = Xi L 0 + Yi L Because Pr(L i0 X R i=i 0+1 c i i 0 + Yi R ) = 1, (17) is equal to c 0 1 i 0. However, (16) is equal to i k=i Pr(L 0+1 k Xk L + Y k 1 L ). So, LHS RHS < 0 because c > c i Case (ii) L i Xi L 0 + Yi L For this case, we have LHS = RHS = c i0. (17) Because the probability of both cases is nonzero, we conclude that (5) does not hold when J 2. Since the multimodular property does not hold for the general case, we do not expect to obtain an optimal schedule without implementing an exhaustive search. These observations motivate us to develop a local search algorithm that is efficient and can be used to obtain schedules with good quality. We present our study on the solution methodology in Section 4. 4 Local Search Algorithm and Sequential Heuristics for Clinical Scheduling Because our scheduling model for heterogeneous patients is not multimodular as shown in Section 3, we first propose a local search algorithm to find good schedules in Section 4.1. The main assumption of the proposed local search algorithm is that the set of patients is known in advance. In many situations, clinics do not know the set of patients that should be scheduled and appointment schedules are generated sequentially along with the patient call-in process. So, in Section 4.2, we extend our study to sequential scheduling and propose two sequential scheduling procedures. 8

9 Local Search Algorithm First, we derive an important property of optimal schedules, that can be used to generate initial schedules. Then, we define the neighborhood of a given schedule and propose dominance rules to reduce the search space in a local search algorithm. Finally, we give the basic steps of the proposed algorithm. Theorem 4 shows that an optimal schedule prefers patients with lower no-show probabilities. Theorem 4. Let S be an optimal schedule of (1). Let j, j 0 J with p j > p j0 and n j, n j0 > 0. If i I S i,j 0 1, then i I S i,j = n j. Proof. Let j, j 0 J with p j > p j0 and n j, n j0 > 0. Let S be an optimal schedule such that for some i 0 I, Si 0,j 0 > 0, and suppose i I S i,j < n j. Let S = S i0,j 0. Then, F( S) F(S ). From the proof of Proposition 1, we have p j0 r p j0 ( i I,i i 0 c i P i (i 0 )) where P i (i 0 ) is the probability of overflow from slot i incurred by adding one patient in slot i 0 to S on the condition of this patient s arrival. Consider the schedule Ŝ = S+ i0,j. Since p j0 r p j0 ( i I,i i 0 c i P i (i 0 )), we have r ( i I,i i 0 c i P i (i 0 )), and thus p j r p j ( i I,i i 0 c i P i (i 0 )). From this, we get p j (r ( i I,i i 0 c i P i (i 0 )) > p j0 (r i I,i i 0 c i P i (i 0 )). Thus, F(Ŝ) > F(S ), a contradiction. Theorem 4 shows that patients with lower no-show probabilities contribute more than those with higher no-show probabilities. We propose a local search algorithm that schedules patients according to their no-show probabilities. Before explaining the local search algorithm in detail, we define the neighborhood of a given schedule. Definition 2. We say schedule S 1 is a neighbor of schedule S 0 if it satisfies the following: (1) S 1 = S 0 ± i0,j 0 for some i 0 I and j 0 J, i.e. S 1 is obtained by adding/removing one patient of type j 0 ; (2) S 1 = S 0 i0,j 0 + i1,j 0 where i 0 i 1, i.e. S 1 is obtained by reassign one patient of type j 0 from slot i 0 to slot i 1 ; (3) S 1 = S 0 i0,j 0 + i1,j 0 i1,j 1 + i0,j 1 where i 0 i 1 and j 0 j 1, i.e. S 1 is obtained by switching the slots for two patients of different no-show probabilities. Note that the size of the neighborhood is O(max{ J, n} 2 ) where n is the number of patients in the schedule. However, in the course of local search, the size of the effective neighborhood can further be reduced as follows: Proposition 5. Let S 0 be a given schedule that is feasible to (1) (i) Assume that S k = S 0 i0,j k + i1,j k F(S 2 ) > F(S 1 ) > F(S 0 ); for k = 1, 2 and p j2 > p j1. If F(S 1 ) > F(S 0 ), then (ii) Assume that S k = S 0 i0,j 0 + i1,j 0 i1,j k + i0,j k for k = 1, 2, and i 0 < i 1. If F(S 1 ) > F(S 0 ) and p j2 > p j1 > p j0, then F(S 2 ) > F(S 1 ) > F(S 0 ); if F(S 1 ) > F(S 0 ) and p j2 < p j1 < p j0, then F(S 2 ) > F(S 1 ) > F(S 0 ). Proof. The is very straightforward to show (i) using the results of Proposition 1 and Theorem 4. Here, we focus on the more difficult proof of (ii). Let x and y be two patients of types j 0 and j 1, respectively. Let S AB denote schedule S on the condition that all patients in A arrive as scheduled and all patients in B are no-shows. The expected profit of schedules S 1 and S 2 are calculated by conditioning the no-show scenarios of patients. F(S 0 ) = (1 p j0 )(1 p j1 )F(S 0 xy) + p j0 (1 p j1 )F(S 0 xy) + (1 p j0 )p j1 F(S 0 xy) + p j0 p j1 F(S 0 xy) 279 F(S 1 ) = (1 p j0 )(1 p j1 )F(S 1 xy) + (1 p j1 )p j0 F(S 1 xy) + (1 p j0 )p j1 F(S 1 xy) + p j0 p j1 F(S 1 xy) 9

10 Note that (S 0 xy) = (S 1 xy) and (S 1 xy) = (S 0 xy). Let S = S 0 i0,j 0 i1,j 1 and Pi (k) be the overflow probability from slot k defined in (8). It can be easily seen that (S 0 xy) = (S + i1,j 1 y) and (S 0 xy) = (S + i0,j 0 x). Similar result holds for S 1. Then, we have the following: F(S 1 ) F(S 0 ) =p j0 (1 p j1 )F(S + i1,j 0 x) + (1 p j0 )p j1 F(S + i0,j 1 y) {p j0 (1 p j1 )F(S + i0,j 0 x) + (1 p i0 )p j1 F(S + i1,j 1 y)} =p j0 (1 p j1 ){F(S + i1,j 0 x) F(S + i0,j 0 x)} + (1 p j0 )p j1 {F(S + i0,j 1 y) F(S + i1,j 1 y)} =p j0 (1 p j1 ){ i I + (1 p j0 )p j1 { i I { = i I c i Pi (i 0 ) c i Pi (i 1 )} i I c i Pi (i 1) c i Pi (i 0)} i I } c i {Pi (i 0 ) Pi (i 1 )} }{p j0 (1 p j1 ) (1 p j0 )p j For the case where p j0 < p j1, we observe that the second term in the last equality is always negative because p j0 < p j1 and 1 p j1 < 1 p j0. Since F(S 1 ) F(S 0 ) > 0, the first term of last equality should be negative. The first term is independent of no-show probabilities of x, y and p j0 (1 p j2 ) (1 p j0 )p j2 < p j0 (1 p j1 ) (1 p j0 )p j1. Therefore, F(S 2 ) > F(S 1 ) > F(S 0 ). Similarly, we can prove the desired results for the case where p j0 > p j1. We observe that the results of Proposition 5 provide guidelines such that local movements can be implemented according to patient no-show probabilities. In particular, using Theorem 4 and Proposition 5, we can obtain better schedules with reduced computational effort. Next, we describe our local search algorithm in detail. For a given schedule S, we define j i = arg max{p j : S i,j 1}, j i = arg min{p j : S i,j 1} and j as the patient type with the lowest no-show probability to be scheduled. From Definition 2, we note that there are four types of neighbors of S obtained by the following local movements: add, remove, reassign and switch, which are numbered by 1,...,4 respectively. Algorithm 1. (1) Initialization: S =. (2) Local Search: For l = 1 to 4 if l = 1: (neighbors obtained by adding) F1 = max{f(s + i,j ) : i I}; if l = 2: (neighbors obtained by removing) F2 = max{f(s i,ji ) : i I}; if l = 3: (neighbors obtained by reassigning) F3 = max{f(s i,j i + k,ji ) : i, k I}; if l = 4: (neighbors obtained by switching) F4 = max{f4 1, F4 2 } where F 1 4 = max{f(s i,j i + k,ji k,jk + i,jk ) : 1 i < k, p ji > p ji }; 301 and F 2 4 = max{f(s i,j i + k,ji k,jk + i,jk ) : 1 i < k, p ji < p ji } end for (3) Find the best schedule, S, in the neighborhood of S, i.e. S = arg max{f l : l = 1,...,4}. 10

11 (4) If F(S ) F(S), update current schedule S = S and go back to Step (2). Otherwise, go to Step (5). (5) Return the local optimal schedule S. In Algorithm 1, the number of neighbors search for a schedule need to be searched is O(max{ 2, n}), which is smaller than the actual neighborhood size. 4.2 Sequential Scheduling Methods As mentioned earlier, many clinics generate schedules in a sequential fashion. Typically, a patient calls requesting an appointment. The scheduler will either add the patient to an existing schedule and give an appointment time or reject the patient. Muthuraman and Lawley [15] propose a myopic scheduling method, which sequentially assigns calling patients to the slot that most increases the expected profit of the resulting schedule. It is called myopic since it does not take the possibility of future call-ins into account when making the current assignment. Patients are added to a schedule until the expected total profit starts decreasing. Although the authors consider heterogeneous patients, their method does not differentiate patients according to their no-show probabilities while generating schedules. However, better schedules can be generated by considering the no-show probabilities of patients. We propose two sequential scheduling algorithms that do this by using the properties explained in Section 4.1. Let S 0 be a fixed schedule for n 1 patients and assume that we need to schedule the n th patient of type j. The patient will be inserted into the schedule if adding this patient increases the objective function value. Corollary 6 shows that the decision of accepting (or rejecting) a patient is independent of the patient s no-show probability. Corollary 6. For the myopic scheduling method in [15], the decision to accept (or reject) the n th patient of type j is independent of p j. Proof. The myopic scheduling method in [15] is as follows. Step 1. Set Si,j 0 = 0 for all i I and j J. Step 2. Wait for k th patient call. Step 3. The k th patient calls in and the patient is of type j 0. Step 4. Compute F(S 0 + i,j0 ) for i I and set i = argmax{f(s 0 + i,j0 ) : i I}. Step 5. If F(S 0 + i,j 0 ) > F(S 0 ), assign the k th patient to slot i and update S 0 = S 0 + i,j 0 and go to Step 2. Otherwise, stop. Assume that the n th patient is assigned to slot i n. Let S = S in,j. From Proposition 1, we have F(S) F(S 0 ) = p j (r 335 i I c ip i (i n )) where P i (i n ) is independent of p j and can be computed without the n th patient. Then, let i n denote the slot index that yields the minimal 336 i I c ip i (i n ). It is clear that if i I c ip i (i 337 n ) r, we can assign patient n to slot i n to increase the expected total profit. Otherwise, patient n will be rejected. Therefore, the decision on the n th 338 patient is independent of p j Based on the results in Proposition 1 and Corollary 6, it is anticipated that overbooking many patients with high no-show probabilities cannot provide the most desirable results. One major drawback of the myopic scheduling method Muthuraman and Lawley [15] is that the objective function depends on the call-in sequence. Different call-in sequences generate schedules which have high variability in the objective function. In order to show the effect of call-in sequences, we consider two sequences. In the first sequence, there are sufficiently many patients of type j 1 before any patient of type j 2 (p j1 > p j2 ). In the second sequence, there are sufficiently many patients of type j 2 before any patient of type j 1. We apply the myopic scheduling algorithm [15] to generate schedules S 1 and S 2, respectively. Clearly, S 1 has patients of type j 1 and S 2 has patients of type j 2. Figure 2 shows F(S1) F(S as a function of p 2) j 1 p j2 for 11

12 Ratio of Expected Total Profit p 1 p 2 Figure 2: Ratio of Expected Total Profit F(S 1 )/F(S 2 ) vs. p j1 p j pairs of randomly generated call-in sequences. The difference between F(S 1 ) and F(S 2 ) increases as p 2 p 1 increases. One may think that it is rare to have all patients at the beginning of the sequence with high no-show probabilities. However, it is commonly observed that patients who make reservations at earlier times tend to have higher no-show probabilities and they often use up the physician s capacity before patients with low no-show probabilities can be scheduled [13, 14, 16]. Therefore, a sequential scheduling method, which accepts all patients regardless of their no-show probabilities, may not generate good schedules in terms of expected profit. In sequential scheduling, there is an existing schedule and the patients who will call-in should be scheduled or rejected. Assume that S 0 is the existing schedule and there are n j patients for all j J that should be scheduled or rejected. The following is the revised model of (1) in the sequential scheduling setting, which adds new patients to an existing schedule: max G(S) =r S i,j p j c i kr i,k i I j J i I k s.t. i I S i,j i I S 0 i,j n j (18) S i,j S 0 i,j 0 S i,j Z i I, j J We give the following result as a corollary of Theorem 4 which can be easily proven using similar argument to that of Theorem 4. Corollary 7. Assume that S is an optimal schedule to (18). For j, j 0 J with p j > p j0, either n j = i I (S i,j S0 i,j ) or i I (S i,j 0 Si,j 0 0 ) = 0, i.e. Si,j 0 = Si,j 0 0 for all i I. Corollary 7 shows that it is always better to include patients of low no-show probabilities into an existing schedule before capacity limit is reached. Also, by Proposition 1 and Corollary 7, our local search algorithm still works for any given existing schedule and patient set. From Corollary 7 and the observation in Figure 2, it is easy to see that limiting the number of patients with high no-show probabilities in the schedule will be an effective way to improve its performance. Following this line, we propose two sequential scheduling methods: the restricted myopic scheduling and the forecasting-based scheduling methods. 12

13 The basic idea of the restricted myopic scheduling method is using upper bounds to restrict the number of patients with high no-show probabilities in the schedule. We set upper bounds, B j, on the number of patients of type j for j 1 in the schedule such that B 1 B 2 B J. Let b j be the number of patients of type j in the current schedule. The basic steps of the restricted myopic scheduling method are as follows: Restricted Myopic Sequential Scheduling Method 1. Step 1. Set b j = 0 and Si,j 0 = 0 for all i I and j J. Step 2. Wait for k th patient call. Step 3. The k th patient call occurs and is of type j 0. Step 4. If b j0 + 1 > B j0, do not accept patient k and go to Step 2. Step 5. Perform the traditional myopic scheduling algorithm to compute the best slot i 0 for patient k. Step 6. If F(S 0 + i0,j 0 ) < F(S 0 ), i.e. adding patient k decreases the expected total profit, stop. Otherwise, update b j0 = b j0 + 1 and S 0 = S 0 + i0,j 0 and go to Step 2. The restricted myopic scheduling method is very simple to implement. However, this method does not consider potential call-ins in the future. We propose another sequential scheduling algorithm, which considers forecasted patient requests from current time to the appointment day. Specifically, for each callin patient, we generate a schedule from the existing schedule considering the forecasted future patients. We believe that the information about anticipated patients contributes to the scheduling algorithm in two ways: (i) this information can limit the number of patients with high no-show probabilities in the final schedule and (ii) slot allocation decisions anticipate possible future patient call-ins. In this study, we simply use average historical data to predict future patient demand. Assume that we are generating forecasted patient demand for a day that is T minutes ahead from current time. We can use the average of q pieces of historical patient demand data that happened T or less minutes before virtual appointment days. We denote the forecasted patient demand by n j for j J. To avoid overestimating future patient arrivals, we may discount our forecasting by α with 0 α 1. When α n j is not an integer, we can round it to the nearest integer. Forecasting-based Sequential Scheduling Method 1. Step 1. Set Si,j 0 = 0 for all i I and j J. Step 2. Wait for k th patient call. Step 3. The k th patient calls in and the patient is of type j 0. Step 4. Predict future patient demand and obtain n j for j J. Step 5. Perform the scheduling algorithm that considers requests α n 1,..., α n j0 1, α n j0 +1, α n j0+1,..., α n J to generate a schedule S from S 0 for (18). Step 6. If i I such that Si,j 0 Si,j 0 0 1, assign the k th patient to slot i and update S 0 = S 0 + i,j0. Step 7. If Si,j 405 S0 i,j = 0 for i I and j J, stop. Otherwise, go to Step 2. When α = 0, the forecasting-based scheduling method reduces to the myopic scheduling method in [15]. Comparing these two sequential scheduling methods, the restricted myopic method is conservative because it mostly considers available patient information while the forecasting based method is aggressive since it heavily uses predicted information on potential patient calls. Note that the successful application of both of them requires that the physician has enough patient demand which is always the case in practice. In Section 5, we perform a computational study to compare the proposed algorithms with the traditional myopic scheduling algorithm in [15]. 13

14 Computational Study We perform a computational study to test the performance of proposed algorithms. We consider three experimental settings. In the first setting, we show the effect of considering heterogeneous patients instead of homogeneous patients. In the second setting, we compare the proposed sequential scheduling algorithms with the traditional myopic scheduling algorithm in [15]. In the last setting, we analyze the effect of overflow cost on expected profit. Throughout our experiments, we assume that a clinic session is 4 hours and partitioned into 8 equal length slots. The service rate λ, which is equal to 2, is constant during the session. Unless explicitly mentioned, r = 100, c i = 40 for i and c = Homogeneous versus Heterogeneous Patients A major contribution of this study is that the variability of no-show rates is taken into consideration while designing the scheduling algorithms. Algorithm 1 is used to schedule heterogeneous patients. The heuristic algorithm proposed by Kaandorp and Koole [7] is used to schedule homogeneous patients. We consider three types of patients. p 2 is set to 0.5, and p 1 and p 3 are randomly generated such that (p 1 + p 3 )/2 = 0.5. We assume equal number of patients in each group (n 1 = n 2 = n 3 = n). We consider different values for n (n = 1,...,12) to analyze the effect of variance on expected total profit for different population sizes. The variance of no-show rates is derived from the variance of p 1, p 2 and p 3. Figure 3 shows the results of 400 randomly generated problems. We observe that the expected profit obtained from the heterogeneous scheduling model dominates the one obtained from the homogeneous scheduling model for all population sizes. The impact of variance of no-show rates on expected profit becomes more significant as the number of patients increases. Figure 4 highlights the difference for 6 and 12 patients. The consideration of heterogeneous patients leads to greater improvements when variance is greater. Especially, when n = 12, the improvement on total expected profit could reach up to 20%. Algorithm 1 schedules more patients with low no-show probabilities. However, the total number of patients scheduled is less. As a consequence, the variance in expected profit is less than the one obtained by homogeneous model Homogeneous Patients Heterogeneous Patients 700 Expected Total Profit Number of Patients (*3) Figure 3: Expected Total Profit vs. Number of Patients Sequential Scheduling We compare the proposed sequential scheduling methods with the traditional myopic scheduling method by Muthuraman and Lawley [15]. We set J = 2, p 1 = 0.8 and p 2 = 0.2. We first randomly generate

15 Homogeneous Patients (6*3) Heterogeneous Patients (6*3) Homogeneous Patients (12*3) Heterogeneous Patients (12*3) Expected Total Profit Variance Figure 4: Impact of Variance for 6 3 and 12 3 Patients 443 call-in sequences that span over 30 days. We assume that the call-in rate increases as the call-in time 444 gets closer to the appointment time. At the beginning, the call-in rate for patients of type j 1 is smaller 445 than the rate for patients of type j 2. As time goes on, it increases and finally becomes larger than that for patients of type j 2. Let λ 1 and λ be the arrival rates of patients of types j 1 and j 2, respectively. Specifically, once a call-in of type j 1 is generated, we update λ = γ λ 1 where γ is a randomly generated positive number that is larger than 1. Similarly, we keep updating λ by a randomly generated number 449 that is less than 1. We generally set parameters in a way such that the number of expected call-ins of 450 both type 1 and 2 throughout mins are more than the number of expected services, λ 8 = 16. In our experiments, we set the initial values λ 1 = and for λ 2 = Random numbers are generated 452 from (1, 1 ± 0.05] respectively. 453 First, we consider the restricted myopic scheduling algorithm. Figure 5 shows the expected total profit 454 for restricted myopic scheduling algorithm (for both B 2 = 4 represented by, and B 2 = 8 represented 455 by ) and the traditional myopic method for 100 randomly generated sequences. The results from the 456 restricted myopic method always dominates those from the traditional myopic scheduling method of [15]. 457 The results from the proposed method are very stable (with less variance), while the traditional myopic 458 method gives results with high variance. As expected, the proposed method obtains better results when 459 the upper bound on the number of patients of type j 2 is lower. 460 To predict potential patient call-ins for a call-in sequence, we randomly choose 4 other sequences to 461 predict numbers of call-ins and set α = 0.5 to discount risk. Figure 6 shows the expected total profit 462 for the forecasting-based sequential scheduling method (represented by ) and the traditional myopic 463 scheduling method for 100 randomly generated call-in sequences. The results show the advantage of 464 using forecasted patient demand to generate schedules sequentially. In cases where many potential 465 patients with low no-show probabilities are expected, it would be wiser to reserve the capacity for these 466 patients rather than adding patients with high no-show probabilities into the schedule. Obviously, this 467 argument justifies the open-access scheduling model in which the capacity is kept until the day before 468 the appointment day or the appointment day since their no-show probabilities are low in those days. 469 In fact, both our theoretical analysis in Sections 2-4 and our computational study on the performance 470 of the restricted myopic scheduling method and the forecasting-based scheduling method show that 471 patients with low no-show probabilities should not be restricted by open access model. Actually, our 472 models show that allowing patients with low no-show probabilities to make appointments ahead is 473 effective. 15

16 Expected Total Profit Sequence ID Figure 5: Restricted Myopic Scheduling Method vs Myopic Scheduling Method Expected Total Profit Sequence ID Figure 6: Forecasting-base Method vs Traditional Myopic Method Cost of Patient Waiting Time The expected revenue and overtime cost can typically be estimated by health care practitioners. However, the value of patient waiting time is determined subjectively. Since the cost associated with patient waiting time is not an actual cost paid by the clinics, it is important to investigate the effect of different cost values on scheduling and expected profit. As discussed in Section 2, patient waiting time is directly related to the size of overflow between slots. So, we control the value of patient waiting time through changing the value of c i for i. Similarly, the cost of physician overtime can be controlled through changing the value of c I. In our experiment, we assume c i = c j if i j for i, j I\{}. We consider two schedules generated by the traditional myopic scheduling method from two calling sequences described in Section 4.2 because of their simple patient structures. For these two sequences, we set p 1 = 0.8 and p 2 = 0.2. Figure 7 shows the effect of c i on expected total profit for both sequences. As overflow cost (c i ) increases, expected total profit decreases for both schedules. However, the expected profit of the schedule for the second call-in sequence decreases faster than that of the schedule for the first sequence. When c i is small (which means that the physician time is more valuable than patient waiting time), the performances of two schedules are close to each other. In such cases, differentiating patients by their no-show probabilities is not very beneficial. When c i is increasing, the difference between the two schedules becomes larger. If the patients waiting time is considered as a significant part of the 16

17 491 performance measure, the number of patients with high no-show probabilities should be restricted Sequence 1 Sequence Expected Total Profit c i Figure 7: Expected Total Profit vs c i Laganga and Lawrence [11] mention that the net overbooking utility, which is the expected net return generated by overbooking, is larger in the case where patients have high no-show probabilities than in the case where patients have low no-show probabilities. They further mention that this phenomenon is more significant when the cost of patient waiting time and physician overtime are high. According to our computational results in Figure 7 and Figure 8, the expected total profit of schedules generated using overbooking decrease when cost of patient waiting time and physician overtime increases. Furthermore, the speed of decrease is faster in the case where patients have higher no-show probabilities. These results indicate that overbooking can compensate the loss from patient no-shows to some extent. However, reducing patients no-show rates should have higher priority than applying overbooking Sequence 1 Sequence Expected Total Profit c I Figure 8: Expected Total Profit vs c I Concluding Remarks and Managerial Insights Various scheduling models with overbooking have been proposed to help health care providers alleviate the negative effects of patient no-shows. However, to the best of our knowledge, all existing studies either assume that patients are homogeneous in terms of their no-show probabilities, or do not consider 17